A C T A U N I V E R S I T A T I S L O D Z I E N S I S FO L IA O E C O N O M IC A 2 1 6 , 20 0 8 ______________ ♦ Wiesław Wagner T E S T S O F M U L T I V A R I A T E N O R M A L I T Y U S I N G S H A P E M E A S U R E S O F T H E D I S T R I B U T I O N
A B S T R A C T . Karl Pearson, in 1990, proposed tw o num erical characteristics o f the distribution o f random variables i.e. asym m etry (sk ew n ess) and kurtosis (flatn ess). Their sam ple approxim ations a llo w to describe partially the em pirical distribution, to find out i f it differs from a sym m etric distribution and i f it is ex ceed in g ly flat or high.
T he m easures o f shape for distributions with know n first four central m om ents are uniquely d efined, in particular, for the univariate normal distribution they are equal to 0 and 3. It a llo w s to com pare distributions w ith know n m easures of shape w ith the nor mal distribution. Such com parisons in univariate ca se is d on e b y m eans o f standardized tests based on the third and fourth sam ple central m om ents. A n o v e r v ie w o f such tests m ay be found in the w ork b y D ’A gostin o and Pearson (1 9 7 3 ).
The translation o f shape m easures to multivariate case w as done by M ardia (1 9 7 0 ). T hese m easures sig n ifica n tly enriched the statistical description o f em pirical distribu tions and allo w ed to introduce m any tests o f multivariate norm ality. T he distributions o f these tests’ statistics usin g sam ple m ultivariate asym m etry and kurtosis are u su ally d e rived through central lim it theorem s.
In the w ork an o v e r v ie w o f m ultivariate norm ality tests based on the sam ple m eas ures o f asym m etry and kurtosis is given . The statistical properties o f these m easures are d iscu ssed as w e ll as the u sefu ln ess o f these tests w ith respect to p ow er and sam ple size.
K ey words: m ultivariate norm ality, statistical tests, shape m easures.
I. D E FIN IT IO N O F M U LTIV A RIA TE RANDOM V E C T O R ’S ASYM M ETRY AND KU RTO SIS
L et X b e /» - d im e n s io n a l r a n d o m v e c to r w ith a d is tr ib u tio n g iv e n b y th e c u m u la tiv e fu n c t io n F p ( x , p , ' L ) = F p , w h e r e x e R p, p is a p - d i m e n s i o n a l v e c to r o f e x p e c te d v a lu e s a n d L , b y a ss u m p tio n , is a p x p - d im e n s io n a l c o v a r ia n c e
matrix positive definite. When the distribution considered is multivariate normal its cumulative distribution function is denoted by N p [ x ,/J ,ľ.) = N p .
Multivariate asymmetry and kurtosis denoted with Д р, ß 2p as random vec tor’s X measures o f shape, are Pearson’s numerical characteristics J ß and ß 2 o f univariate case, generalized to p-dimensional case. The numerical character istics ß lp, ß 2p are defined with a form
(a) bilinear
P „ =
e[
[ ( x - ^ z - ' ( J r . - x ) ] ’ }for independent /7-dimensional random vectors X , X . identically distributed,
(b) quadratic
Pl r = Ą [ ( x - r t r ' ( x . - M)] J
for /»-dimensional random vector X, where S '1 is the inverse matrix o f I . In particular, for p = 1 , we have /?,, = Д and ß 2l = ß 2 .
The properties of P \p , ß 2p when X ~ N p are given by two lemmas. Lem m a 1. If X ~ N p then ß lp = 0 .
Proof. Let Y = £ “|/2 (ЛТ ~ / j )-T h en Y ~ N p { 0 , l ) , w h e r e /is a unit ma trix and ß Xp = is j (Y' Y. )3 ] = 0, because ordinary moments o f odd order o f the
standard normal distribution are equal to zero and each component o f two-linear form Y' Y, contains at least one variable in odd power.
Lem m a 2. If X ~ N p then ß lp = p ( p + 2) = g ■
Proof. Let us make use o f the formulae for the expectation and variance of the quadratic form
E ( X ' A X ) = E ( X ' ) A E ( X ) + t r ( l A ) ,
where //•(•) stands for matrix trace. After substitutions X —> ( X - / / ) and
A - » I ' 1, we get
P 2p = e [ [ ( ^ - ^ ) ' i - , ( x . - / / ) ] 2} = d í ( [ ( ^ - ^ ) - z - ' ( ^ - ^ ) ] 2) +
+ { е [ ( Х - И ) " £ Г ' ( Х . - И)] }2 = 2 p + p 1 = /?(/? + 2).
When p = 2 , i.e. for twovariate distributions, we have
ß a = ( 1 - P 2)"3 {у]о + /оз + 3(1 + 2 р г )(у]г + r l ) - 2Л 3змз + + b p { y i0( p y n - Г 2 \ ) + Г<а(р/21 - Г п) - ( 2 + Р2)УпГ,г]}>
ßll
= [^40 + /о4 + 2 у22 ■*" (^22 — У|3 — У з | ) ] ^ ~ Z7 ) 1 where X = ( X ]tX 2j , р = {м„рг), o f = D 2( X {), er] = D(X2), p = C orr(X ,,X 2), r„ = / ^ / « 20 , д , = E {(X ,-A )r(X2- ^ ) ‘}In particular, when p = 0 , i.e. when the coordinates o f the two dimensional random vector are independent, then
ß \ 2 = ľ lo + /0 3 + 3 ( r .2 + Гг2. ) a n d ß l 2 = ^40 + Г<)4 + 2 Г22 •
Now we will prove lemmas 1 and 2 in the case of p = 2 , i.e. for the two dimensional normal distribution.
Lemma 3. Д 2 = O a n d /? 22 = 8 if ( X ltX 2) ' ~ N 2.
Proof: We apply the well known formulae for distribution’s moments (see e.g. Kendall and Stuart 1963 p. 91 ):
yn
=y2\
= /3 0 = 0 from which imme diately follows that ß n = 0. To prove ß 22 we apply other known formulaeУ4o = r,M = 3,
y3
l
=yl
3
= 3 andy22
= 1 + 2p 2, from where we haveß 22 = {3 + 3 + 2(1 + 2/>2) + 4/>2[1 + 2/>2 - 3 - 3 ] } / ( 1 - / > 2)2 =
The formulae for ß n and ß n for some two dimensional distributions are given in the form o f (Mardia 1974, Mardia et al. 1979, Davis 1980):
a) the mixture o f two dimensional normal distributions
h ( x ) = 0 , 8 f ( x , 0 , 1) + 0 ,2 f ( x ,0 , cr21 ) , where / ( • ) is the density function
and ß n = 0, ß n = 8
(er2 + 4 )2
b) two dimensional gamma distribution
for (x,,x2) ' 6 R 2 = (0 ,о о )д г (0 ,о о )-Д 2 = 0 , 36
ß n = 11>
c) two dimensional exponential distribution
P ( X { > x{, X 2 > х г) - exp[-x, - x 2 - max(x,, x2)],
(дг,х2) е Л 2; ß n = З/з4 + 9 p ł +15/92 + 1 2 p + 42 \3 2(1 - p )
5 + p - p 1 - 3 p 3
ß n 4(1- p 2) ’
d) twodimensional Morgenstern distribution
/*(*) = 1 + 3/0(1 - 2x,)(1 - 2x2), X6 (0,1) x (0,1), ß n = 0 ,
ß n = 4(7 - 1 Ъ р1) /(5(1 - p 1) 2.
II. M U LT IV A R IA TE SA M PLE ASYM M ETRY AND K U RTO SIS
The estimators blp and b2p o f ß lp and ß lp based onp-dim ensional sam ple U = { X x, X 2, . . . , X n) are expressed through the powers o f bilinear and quadratic form (Mardia 1970, 1974, 1977) in the following way:
b) sample multivariate kurtosis (flatness)
b2p = - t s 2^ n j .I
where g tj, = ( X j S~l ( X ľ ~ X ) and X and S are the mean vector and
covariance matrix based on sample U. The forms g j . and g M may also be expressed through scaled residual vectors Yj = S~U2( X j - X ) , then
g ir = Y'j Yj, and gjj = Y j'Yj . In this notation 11 stands for the inverse
matrix S ' 12, so that S ' 12 ( S ' 12)’ = S.
The random variables b ip and b2p have distributions implied the distribu tion o f the random vector X whose independent realizations are expressed by matrix U. For these variables the distribution characteristics in the case o f the multivariate nonnal distribution are the following (Mardia 1970, 1974, 1977, Mardia and Kazanawa 1983, Mardia and Foster 1983):
a) g((n + l)(p + l) 6^ g = p(p + 2), (л + 1)(и + 3) b) D 2(bip) = ]2('P + ^ P + 21, c) d) E(blp) =g ( n - l ) и + 1 ’ D \ b lp) =8g(n - 3)(n - p - l)(w - p + 1) (л + 1)2(и + 3)(и + 5) e) n 2 g ( 9 p 4 + \ 2 p i - \ 9 2 p 2 — 328/? + 256
П (h 4 - 64g (P + 8) O /^3 > 2 p) 2 * (и + 1)2(« + 3)(« + 5) (л - 3)(/i - p - l)(n - p +1) h) Cov(bt ,Ьг ) = 12p h / n 2, h - 8p2 -13/? + 23, i) Corr(bip,b2p) = 3/i i , r . k) o ' ( » 6 / ~ l2 v ’ ( / ) . [6я(р + 1)(р + 2)2]|/2 ’ '/> + 2 ч ŕ f + i v 3 ,, 4 0 = ■ , r 1) CoV(s[b^p ,b2p) = - ^ L nyjnf m) Д Ц / 5 > >/2v (/)(1 - 2 / + 4v2 ( / ) ) 2 / r \ \ 3 / 2 n) Co r r( Jb ~ b2p) = yj6ph n j ň f 8 g ( 6 / - 1 2 v29 / ) 1/2
The above given formulea for the moments o f random variables bXp and
b2 are correct for big n, and their approximations were given up to the order o f
III. TESTS OF MULTIVARIATE NORMALITY BASED ON bXp AND b2p
For the distribution N p we have ß \ p = ®< anc* ß i p = P(P + 2) = g. In vestigating the /»-dimensional empirical distribution Pp by means o f independ ent observable random vectors X itX 2, . . . , X n we ask if they come from a mul tivariate population with ß ip = 0 or A , - * or, simoultaneously, ß lp = 0 and
ß 2p - g. This leads to define the null hypothesis as H 0 : Pp e N p against the
alternative hypothesis determined by one of the distribution classes:
Ą - f l t p * 0 , ß 2p= g l A1 - ß {p=o, ß l p * g ; A i - ß l p * o , ß 2p* g .
There are many tests o f multivariate normality to verify the hypothesis for mulated which are based on sample statistics blp and b2p. We will differentiate between the omnibus and directed tests.
Definition 1. Statistical tests for a determined class o f alternative distribu tions will be called directed tests.
Definition 2. Statistical tests most powerful in the class o f possible alterna tive distributions will be called omnibus tests.
To verify the null hypothesis against # , : Pp e A, or / / , : Pp e A2 we use the tests o f multivariate normality based on the tests statistics being the equiva lents o f b, l p and b, . The directed tests used will be most powerful for distribu-2 p tion classes A, and A2. For the omnibus tests and the alternative defined by family A } we apply the tests’ statistics being functions blp or b2p. These tests have the property o f simoultaneous assessment o f the departures of multivariate assymetry and kurtosis o f the empirical distribution Pp from ß Xp = 0 and
ß 2p = g. These tests are recommended whenever we do not have any prior in
The descriptive statistics based on blp and b2p have distributions known for big n, which are given by lemmas 4 and 5.
i ( p + A
Lem m a 4. nbip/ 6 ~ f = \ I, when U ~ M N p that is when U has
the matrix nonnal distribution (see e.g. Wagner 1990). Lem m a 5. (b2p - E ( b 2p) ) l D[b2p) ~ N( 0, l ).
The proofs o f the lemmas may be found in Mardia (1970, 1974), as well as in Domański W agner (1982).
From the applicational point o f view, one differentiates between the tests of multivariate normality based on bXp and blp with respect to the determined class o f alternative distributions, in the following way:
• Ai - M „ C „ L ( b {p) , u { b yp) , w { b [p),Q, tests;
• A2- M 2,C2, U ( b 2p) , W [ b 2p) ,Q2 tests;
• A3- M i ,C3,CĄ, S l , S j l , S ^ , C l , C 2R,Q tests.
In what folows, we review the above mentioned tests, limiting ourselves to mentioning: (1) author (or authors), (2) test’s statistic and (3) the distribution of the test’s statistic, always assuming that U ~ M N .
A. Tests for hypothesis / / „ : Pp e N p or # , : Pp e Ar .
(a) (1) Mardia (1970); (2) M, = nbip /6 ; (3) (lemma 4);
(b) (2) Bera i John (1983); (2) C, = , (3) X 2. ,
0 1=1
T, = £ у , 2 / я , 1 = 1 ,2 .
Yy = •S'"1/2(Arj - X ) = ( Y ]j, Y 2j, . . ., Ypjy ;
(c) (1) Mardia and Foster (1983); (2) L( blp) = у + S \ n ( b lp -<!;), (3) N(0,1) y , S , ^ - SL Johnson distribution parameters;
(d) (1) Mardia and Foster (1983); (2) U ( b . ) =— К ~ 6 f / nP
6 ^ 2 f i n 2
; (3) N(0,1);
1
[ š * N
- 3 / + 3
(e) (1) Mardia and Foster (1983); (2) W(bXp) = \ —n f ib[p - 3 / + — l ^ j l f
(the Wilson-Hilferty approximation of distribution bip ); (3) N (0,1).
B. Tests for hypothesis H 0 : Pp e N p or H x : Pp e A 2 .
(f) (1) Mardia (1970); (2) M 2 = ^ ; (3) Zi'> 8 g / n
(g) (1) Bera and John (1983); (2) C2 = — « £ ( Г . - 3 > ’ + £ ( 7 - ,.- l) !
/ - I I Z i < ľ i p
(3) T . - Y X l n , r » = X ( W 1 / n - r , as in (b);
y=l У-1
6, - g ( n - l ) / ( / j + l) (h) (1) Mardia and Foster (1983); (2) U(b2p) =
-y ß g / n 2
(3) N(0,1);
(i) (1) Mardia i Foster (1983);
(2) W(b2B) = з Д 1 ~ - 1 - 2 /9 / ;
2/J V
2 I 9 /; l + a p / ( f - 4 ) \; (3) N(0,1);
/• < + 2 ) _ b 2P - E ( b 2p )
f = 6 + 4 ( d + J d + d ) , d - 2(/7 + 8)2 ’ D ( ^ } •
C. Tests for hypothesis 7 /0 : Pp £ N p or H x \ Pp e А ъ.
(j) (1) Jargune and Mckenzie (1982); (2) Л /3 = Л /, + Л / 2; ( 3) ,£y+1;
4 É ^ 2 + É ( ^ - 3) (к) (1) Вега and John (1983); (2)
i=i
(m )(l) Mardia and Foster (1983); (2) S 2Ĺ = L2 p) + U 2(b2p) , S l = U \ b ip) + U 2(blp),\ p f ' ^ v 2p /» S 2 = W \ b J + W \ b 2l>),r t \ L' \ p J 1 rr У У 2 p > “li y^»2 N — U r O , C 2n =b 'V~' b, C l = d ' V ~ ' d , b = h i p - 6 / / Л , v ="1 2 f t n 2 \ 2 p h / n 2 b2p- g ( n - l ) / ( n + l ) \ 2 p h ! n 2 8 g i n d = ’y[bTP - E ( J b ~ j n - 1 И+1
(3) all statistics mentioned here follow the chi-square distribution with two degrees o f freedom;
(n) Small (1980); (2) £ , = « Y (1), P , e A , ; Q 2 = Y (2)U ^ Y(2), P , e A 2 0 = 0 , + 0 2 , P p 6 A 3, Y 0> = y2 = <5* s in h '1 (y]b{ ( X { ) / Л,* S ' s i n h ' 1 Q b ^ X ^ / ^ r 2 + S1s w h - l[(b2( X l) - 4 2) / Ä 1] y2 + iJ2sin h-,[(62( X J - ^ ) / A 2] Ц|) = (»■«■)» U (2) = ( 4 ) . i’ = 1.2,...,A
where riľ are sample coefficients o f linear correlation determined from matrix U. The mentioned constants 5*\ and \ / X \ can be found in the tables given by D ’Agostino and Pearson (1973). The remaining constants S 2, y 2, Л2, £ 2 are determined according to the principle o f parameter estimation in the S l Johnson distribution family.
IV. T H E P R O P E R T IE S O F M U LT IV A R IA TE N O R M A LITY TESTS USING blp AND b2p STA TISTIC S
In chapter 3 we mentioned many tests o f multivariate normality o f the di rected and omnibus type. Some o f them have simple form o f tests’ statistics, other require additional numerical calculations.
More important properties o f multivariate tests o f normality using b]p and
b2p statistics are as follows:
a) they use scaled vectors o f residuals allowing to find big residuals when big were the bXp and b2p statistics;
b) they use the numerical characteristics o f the distribution o f bip and К sta tistics mentioned in chapter 3;
c) they make use o f constant parameters o f the Johnson’s family o f distribu tions which are determined by means o f special numerical methods;
d) they have the limiting distribution either normal or chi-square; e) the chi-square degrees o f freedom depend only on p;
0 tests are appropriate for big n, because the numerical characteristics o f blp and blp were given for order 0( ri 2);
g) the omnibus tests using both blp and will be good for applications in which the correlation between random variables blp and b2p is sufficiently small i.e. whenever there the condition is n > 300 (8 p 2 - 1 Ър + 23) / ( p +1) met;
h) the Wilson-Hilferty transformation for constructing tests based on bip may replaced with a transformation corrected by Goldstein (1973);
i) because bip and b2p are invariant with respect to affine transformations, the tests based on bip and b2p possess same property ;
j) there are no major numerical difficulties in determining tests statistics when one assumes that the sample covariance matrix is positive defined;
k) tests based on bXp and b2p have moderate power for undetermined alterna tive distributions and the power is higher for directed tests.
To illustrate the strength o f correlation between bip and b2p influencing the usefulness o f some tests o f multivariate normality, in table 1 we present the re sults of our own computations:
Table I Correlation between blpand Ьг/1
.. p _____ n Corr ľ n Corr 1 30 0.9487 4 30 1.6499 50 0.7348 50 1.2781 100 0.5164 100 0.9087 200 0.3674 200 0.6390 2 30 0.9360 5 30 2.0605 50 0.7250 50 1,5960 100 0.5127 100 1.1285 200 0.3625 200 0.7980 3 30 1.2522 6 30 0.9533 50 0.9699 50 0.7784 100 0.6859 100 0.6029 200 0.4649 200 0.4767
The above presented calculations prove how important it is to determine the proper sample size with respect to the number o f variables investigated p in order to get a reasonable correlation measure. The general coclusion is that this number grows with the number o f variables.
V. CO NCLUSIO NS
We gave an overview o f more important tests o f multivariate normality based on statistics bip and b2p, corresponding to multivariate sample asymmetry and curtosis. These tests were also extensively discussed by K. V. Mardia in the seventies and eighties.
The tests mentioned are characterised with good power, especially in the case o f directed tests. Many tetsts use the sample vector o f means and sample covariance matrix which is assumed to be positive defined. These assumptions may be weakened, then, instead o f normal inversion the so called g-inversion is applied. This makes the scope o f practical applications o f the tests discussed, much wider.
Investigating tests using statistics bip and b2p has been slightly curtailed, mainly, due to the fact that more powerful tests o f multivariate normality have been proposed (e.g. those based on stochastic processes or empirical charteristic functions).
A general overview o f other tests o f multivariate normality based on ran domization principle, union and Roy’s intersection, power transformation as well as radii and angles with the use o f multivariate geometry was given in a monograph by Domański et al. (1998). On the other hand, W agner (1990), gives a generalized Shapiro-Wilk test.
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T E ST Y W IE L O W Y M IA R O W E J N O R M A LN O ŚC I K O R Z Y S T A JĄ C E Z M IA R K SZTA ŁTU RO ZK LA D U
M iary kształtu rozkładu jed n o - i w ielow ym iarow ych zm ien n ych lo so w y c h znajdują p ow szech n e za sto so w a n ie w konstrukcji testów jed n o- i w ielow ym iarow ej norm alności. Przy ich konstrukcji korzysta się z p ierw szych czterech m om en tów centralnych w yp ro w adzanych z o d p o w ied n ich statystyk próbkow ych przy o d p ow ied n ich założeniach sto chastycznych.
W pracy dokonano przeglądu testów w ielow ym iarow ej norm alności opartych na próbkow ych m iarach asym etrii i kurtozy. Podano różne ich w ła sn o ści statystyczn e, u w zględ n iające w ielk o ści prób oraz postacie p rzekształcone d o jed n ow ym iarow ych statystyk próbkow ych. Z ałączon e zostały rów nież w yniki badań d o ty czą ce m o c y testów .
